Texture only one side of face
In 3D view, Tab into Edit Mode. Set your selection mode to face select (mode toggle is in the upper left hand corner of the 3D viewport). Select the inside faces of the vase. Press P to separate then select Separate by Selection from the pop-up menu. It should be the default option. If so you can just left click after pressing P. Tab into Object Mode, select the separated inner surfaces and apply a new material to them.